A Rigidity Program for Eliminating Density–0 Exceptions in the Riemann Hypothesis

## Overview This record contains the v5. 0 main closure paper of the RH/GRH rigidity program. The paper develops a renormalized rigidity framework for \ (L\) -functions whose purpose is to eliminate density-0 exceptional windows and to close the analytic chain from explicit formulae to Wasserstein \ (W₂\) displacement convexity. The v5. 0 release is explicitly modular. - **RH Circle A v5. 0** provides the rigorous smoothed / density-1 core, together with the distortion-stable \ (R\) -circle reduction. - **Density-0 Proof v5. 0** provides the two closure gates that are not claimed inside Circle A itself: - **Gate (A): ** density-1 / log-density-1 control \ (\) exception-free control on all sufficiently large windows; - **Gate (B): ** remainder \ (\) metric slope \ (\) rigidity energy for each fixed admissible target. - **RH Guide v5. 0** is a reader-facing navigation document with dependency tables and jump ledgers. - **Selberg Class Extension v5. 0** is an optional extension layer and is not part of the verified degree-\ (2\) closure spine. In addition, the Gate (A) step is now separately archived as a standalone reviewer-facing audit note: - **Gate (A) standalone audit note v1. 1** (DOI: **10. 5281/zenodo. 18922927**) isolates the dyadic-spike elimination mechanism that upgrades density-1 / registry control to an all-sufficiently-large-windows conclusion. ## Verified low-degree scope Within the declared admissible parameter box, the verified scope of the v5. 0 release is degree \ (2\): - the Riemann zeta function;- primitive Dirichlet \ (L\) -functions;- holomorphic \ (GL (2) \) newforms. In this regime, the closure is organized into three theorem layers: 1. **H1/H2/H3 package** - well-posedness of the renormalized rigidity functional; - \ (W₂\) -geodesic displacement convexity; - explicit-formula plus large-sieve cancellation, together with the pointwise upgrade package; 2. **Gate (A): all-window upgrade** - elimination of density-0 exceptional windows at large height; 3. **Gate (B): fixed-target closure** - remainder \ (\) slope \ (\) energy vanishing for each admissible target. According to the theorem-level closure package presented in the paper, this yields: - **RH for \ ( (s) \) **;- **GRH for primitive Dirichlet \ (L\) -functions and holomorphic \ (GL (2) \) newforms**. Equivalently, the v5. 0 release presents a theorem-level RH/GRH rigidity closure package in verified degree \ (2\). ## What is new in v5. 0 Relative to earlier versions, v5. 0 emphasizes a cleaner and more explicit closure architecture: - a front-facing separation between the Circle A density-1 core and the imported all-window / fixed-target closure;- a theorem-level **Gate (A) / (B) ** package for density-0 elimination and the chain \ remainder \;\; metric slope \;\; rigidity energy; \- a reader-facing dependency / jump ledger designed to reduce common misreadings;- a sharper separation between the verified low-degree layer and the optional higher-degree extension agenda;- a separately archived **Gate (A) standalone audit note** that extracts the density-1 / registry-control \ (\) all-sufficiently-large-windows step as an independent audit target. ## Standalone audit note The v5. 0 record is now complemented by a separate Gate (A) audit document: - **Gate (A) standalone audit note v1. 1**- DOI: **10. 5281/zenodo. 18922927** This standalone note does not replace the main closure paper. Instead, it is intended to reduce reviewer burden by isolating one specific load-bearing jump from the full package: -1 windowwise / registry control\;\;all sufficiently large admissible windows. \ It is best read as a focused audit companion to the present v5. 0 main record, not as an independent fixed-target RH closure paper. ## Higher-degree Selberg extension Any extension beyond degree \ (2\) is intentionally separated from the proof spine of the present record. For higher-degree Selberg-class families, the remaining gap is reduced to explicit family-uniform inputs, including: 1. uniform Archimedean curvature lower bounds for the relevant gamma-factor geometry;2. family-uniform large-sieve constants at the explicit-formula band limit. These higher-degree requirements are isolated as an **optional extension layer** and do **not** enter the verified degree-\ (2\) RH/GRH closure. ## Optional research modules The v5. 0 release also contains optional research modules that are not part of the verified degree-\ (2\) closure chain. In particular: - the Selberg-class extension note is modular and optional;- the MUGS-oriented appendices in the main paper are retained only as optional proof-skeleton / interface material;- finite-height verification or numerical validation is logically separate from the analytic large-height closure. ## Companion documents This record is intended to be read together with the companion papers: - `Density-0Proofᵥ5. 0`- `RHCircleAᵥ5. 0`- `RHGuideᵥ5. 0`- `SelbergClassExtensionᵥ5. 0` (optional extension layer) - `GateAₛtandaloneᵣefinedᵥ1. 1` (standalone Gate (A) audit note; DOI: **10. 5281/zenodo. 18922927**) ## Summary The v5. 0 release should be read as a modular theorem-level rigidity package whose verified scope is degree \ (2\), together with a sharply delimited optional Selberg-class extension agenda and a separately archived standalone audit note for Gate (A). ==================================== Author: Byoungwoo Lee (leeclinic@protonmail. com / Daejeon, South Korea)